18 W. G. BADE, H. G. DALES, Z. A. LYKOVA
where VJ E is the symmetric tensor product of j copies of E, identified as the subspace
of symmetric elements in (g)3 E, and the product is specified by
(Ew»)v(E^)
= E E
u^vi-
k i-\-j=k
Now suppose that E is a Banach space. The closure of yJ E m ® -^ is denoted by
^ j . 3
\/ E; it consists of the symmetric elements in (g) E. The projective symmetric algebra
of E is
V^=|(t*
i
):ti
j
€\/£? 0'€Z + )1,
with the product
\i+j=k J
this algebra is a commutative, unital Frechet algebra. We consider the Banach subalgebra
BE of \JE defined by
BE = lu = (Uj) e \/E : ||n|| = £ \\u01|. oo i
It is claimed in [Ya] that BE is always semisimple. In fact, this holds only in the case
where E has the approximation property, for one needs to know that, for each fcEN,
the linear functional in (E')k separate the points of V ^ - (If one replaces the projective
tensor norm by the injective tensor norm throughout, the algebra analogous to BE is
semisimple for each Banach space E.) The details of the above remarks are given in
[DaMc3] and [Le].
We can now present Yakovlev's example.
1.13 THEOREM. There is a semisimple, unital Banach algebra that has a singular, com-
mutative extension such that the extension is not admissible, but such that the extension
splits algebraically.
PROOF: We first present a non-commutative Banach algebra with all the other required
properties.
Let
a : 0 E - U F -UG—0
be a short exact sequence of Banach spaces, where i and q are bounded linear operators.
First form AG , a unital, semisimple Banach algebra.
Next let 21 be
CxF x FJ(g) G
k=2
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